Definition 2.7 ([[1]}, Definition 3.10, Remark 3.11 and Definition 3.12]) Let
\(\mathcal {F}\)  be a coherent sheaf on \(\mathcal {X}\) . The modified Hilbert polynomial of \(\mathcal {F}\)  is defined as
\(P_{\mathcal {E}}(\mathcal {F})(m):=\chi (\mathcal {X},\mathcal {F}\otimes \mathcal {E}^\vee \otimes \pi ^*\mathcal {O}_{X}(m))=\chi (X,F_{\mathcal {E}}(\mathcal {F})(m))=P(F_{\mathcal {E}}(\mathcal {F})(m)).\) 
